Problem:
Find the sum of all the roots, real and non-real, of the equation x2001+(21−x)2001=0, given that there are no multiple roots.
Solution:
Apply the binomial theorem to write
0=x2001+(21−x)2001=x2001−(x−21)2001=x2001−x2001+2001⋅x2000(21)−22001⋅2000x1999(21)2+⋯=22001x2000−2001⋅250x1999+⋯
The formula for the sum of the roots yields 2001⋅250⋅20012=500.
The problems on this page are the property of the MAA's American Mathematics Competitions